Related papers: On Exact Learning of $d$-Monotone Functions

We address the problem of finding optimal strategies for computing Boolean symmetric functions. We consider a collocated network, where each node's transmissions can be heard by every other node. Each node has a Boolean measurement and we…

We prove that any submodular function f: {0,1}^n -> {0,1,...,k} can be represented as a pseudo-Boolean 2k-DNF formula. Pseudo-Boolean DNFs are a natural generalization of DNF representation for functions with integer range. Each term in…

The study of monotone Boolean functions (MBFs) has a long history. We explore a connection between MBFs and ordinary differential equation (ODE) models of gene regulation, and, in particular, a problem of the realization of an MBF as a…

Binary classification from positive-only samples is a variant of PAC learning where the learner receives i.i.d. positive samples and aims to learn a classifier with low error. Previous work by Natarajan, Gereb-Graus, and Shvaytser…

We investigate the computational efficiency of multitask learning of Boolean functions over the $d$-dimensional hypercube, that are related by means of a feature representation of size $k \ll d$ shared across all tasks. We present a…

We give improved and almost optimal testers for several classes of Boolean functions on $n$ inputs that have concise representation in the uniform and distribution-free model. Classes, such as $k$-junta, $k$-linear functions, $s$-term DNF,…

We study approximation of Boolean functions by low-degree polynomials over the ring $\mathbb{Z}/2^k\mathbb{Z}$. More precisely, given a Boolean function $F:\{0,1\}^n \rightarrow \{0,1\}$, define its $k$-lift to be $F_k:\{0,1\}^n \rightarrow…

We describe a $\tilde{O}(d^{5/6})$-query monotonicity tester for Boolean functions $f:[n]^d \to \{0,1\}$ on the $n$-hypergrid. This is the first $o(d)$ monotonicity tester with query complexity independent of $n$. Motivated by this…

We develop a method for training neural networks on Boolean data in which the values at all nodes are strictly $\pm 1$, and the resulting models are typically equivalent to networks whose nonzero weights are also $\pm 1$. The method…

We are interested in representations and characterizations of lattice polynomial functions f:L^n -> L, where L is a given bounded distributive lattice. In companion papers [arXiv 0901.4888, arXiv 0808.2619], we investigated certain…

Let ${\mathbb T}=({\bf T},\leq)$ and ${\mathbb T}_{1}=({\bf T}_{1},\leq_{1})$ be linearly ordered sets and $\mathscr{X}$ be a topological space. The main result of the paper is the following: If function $\boldsymbol{f}(t,x):{\bf…

In this article we investigate the property of complete monotonicity within a special family $\mathcal{F}_s$ of functions in $s$ variables involving logarithms. The main result of this work provides a linear isomorphism between…

A monotone Boolean (OR,AND) circuit computing a monotone Boolean function f is a read-k circuit if the polynomial produced (purely syntactically) by the arithmetic (+,x) version of the circuit has the property that for every prime implicant…

We study the problem of estimating a monotone function $f:\{0,1\}^d\to[0,1]$ from noisy observations at uniformly random vertices of the Boolean hypercube. As a measure of complexity for the target~$f$, we use the total $L^1$-influence…

The multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models. Given a sequence length $k$, attention matrices…

We study the problem of approximating and learning coverage functions. A function $c: 2^{[n]} \rightarrow \mathbf{R}^{+}$ is a coverage function, if there exists a universe $U$ with non-negative weights $w(u)$ for each $u \in U$ and subsets…

We study the fundamental problem of learning a single neuron, i.e., a function of the form $\mathbf{x}\mapsto\sigma(\mathbf{w}\cdot\mathbf{x})$ for monotone activations $\sigma:\mathbb{R}\mapsto\mathbb{R}$, with respect to the $L_2^2$-loss…

In this paper we study the approximate learnability of valuations commonly used throughout economics and game theory for the quantitative encoding of agent preferences. We provide upper and lower bounds regarding the learnability of…

We prove that almost every level set of a Sobolev function in a planar domain consists of points, Jordan curves, or homeomorphic copies of an interval. For monotone Sobolev functions in the plane we have the stronger conclusion that almost…

The subject of this textbook is the analysis of Boolean functions. Roughly speaking, this refers to studying Boolean functions $f : \{0,1\}^n \to \{0,1\}$ via their Fourier expansion and other analytic means. Boolean functions are perhaps…